A Balanced Bracket Redux

In one of my earliest experiments, I tested one of the “balanced brackets” drawn by Joe Czapski (available from his website tournamentdesign.org). In this post, I’ll revisit that analysis using the better simulator I have now, which ought to give Joe’s creation fairer (in both the fairness (B) and (C) senses) consideration.

Joe has drawn a large number of brackets for double-, triple-, and even quadruple-elimination tourneys. And he’s done so in very creative ways. His guiding objective is to equalize the number of wins needed to win the tourney as a whole. So, for example, the winner of his 16DE will have exactly six wins (and, possibly, one loss), no matter when the one loss happened. In an ordinary bracket, the path of a champion who stays in the winner’s bracket requires only five wins. But the road to redemption for a player who loses in the first round requires eight wins to fight back through the loser’s bracket and a recharge round (or seven if the bracket is shifted).

Balancing the bracket in this way sometimes requires the pairing of players with different won-loss records, and “if necessary” matches that are played or not depending on the result of those unbalanced matches.

As before, I’ll analyze his 2e16p bracket, and compare its results against other 16DE brackets. Here is my redrawing, with a few non-substantive changes to render it in tourneygeek style: jc2e16p. Note that this differs from the one I analyzed before, which omitted the recharge round. As the recharge was specified by Joe’s original bracket (and as I now have a simulator that’s capable of handling one), I’ve restored the recharge.

In the earlier result, Joe’s bracket faired poorly chiefly because individual rounds were unbalanced. This is only to be expected, of course, from rounds like round I, which pits an undefeated player agains one who’s got a loss. But other rounds are also unbalanced because they do not allow for skill progression.

The early version of the simulator used in the previous analysis had no parameter to adjust the balance of skill and luck – that balance was set at a level I now know is quite high-skill. This is important because that means that the degree of skill progression was also high. But analyses of other brackets that have round-level inequities based on skill progression has shown that the resulting (relative) loss of fairness caused by the imbalance diminishes as the effect of luck increases. That suggests that Joe’s balanced bracket should to better when tested at higher levels of luck.

Here are the results of the test (all unlabeled numbers represent fairness (C)):

with recharge

without recharge

Luck = 1

Luck = 3

Luck = 1

Luck = 3

JC Balance

15.86

68.45

19.49

72.70

Unshifted

16.34

71.69

19.24

73.71

Shifted

16.34

71.17

19.12

72.43

It turns out the Joe’s bracket outperforms both the shifted and the unshifted 16 brackets, even at the lower luck value.

So, how is it that the earlier result showed the balanced bracket coming in last? It turns out that the recharge round is really essential to the whole idea of a balanced bracket. Re-running the experiment without the recharge, Joe’s bracket runs last when luck is low, and in the middle when luck is high. That’s the result I would have predicted at the outset.

With the recharge, the balanced bracket comes into its own. Here is the full analyzed bracket for Joe’s 2e16p at luck = 1: jc2e16panal.

The original analysis was run with a version of the simulator that simply wasn’t up to the job, and as a result unfairly denigrated the balanced bracket. I regret this, and am glad to be able to make what amends I can now.

This does not mean, necessarily, that I now consider the balanced bracket to be best practice for running a 16 double-elimination tourney. There are still troubling local imbalances, as some eye-wateringly high f(b:X) values on the analyzed bracket attest. It’s still true, as I found earlier, that the balanced bracket causes a greater number of repeated matches. The contingent J1 match is so unfamiliar that it’s bound to be misunderstood and improperly administered. If nothing else, I can comfortably predict that any director who adopts such a bracket will face a litany of fairness (A) objections.

But if it would be a mistake to embrace the balanced bracket without reservation, it would be an even greater mistake to reject it out of hand. In addition to the fairness (C) advantages, there are some nice flow characteristics to the balanced bracket.

For now, let’s welcome the balanced bracket as a worthy contender. No doubt we’ll learn more about its merits and defects.

Another interesting thing about these “balanced” brackets is that they can still work well for numbers that are not powers of 2. I’d like to see a comparison of the 14- or 18-team brackets to a “normal” structure.